How to simplify fractions
Simplifying a fraction means rewriting it with the smallest possible numerator and denominator while keeping the same value. To do this, find the greatest common divisor (GCD) of the two numbers and divide both by it. A fraction is fully simplified once the numerator and denominator share no common divisor other than 1.
Formulas used
Simplification: simplified fraction = (numerator ÷ GCD) / (denominator ÷ GCD)
Addition: a/b + c/d = (a×d + c×b) / (b×d)
Subtraction: a/b − c/d = (a×d − c×b) / (b×d)
Multiplication: a/b × c/d = (a×c) / (b×d)
Division: a/b ÷ c/d = (a×d) / (b×c)
Worked example
To add 3/4 and 1/6, first find a common denominator by multiplying 4 by 6, which gives 24. The calculation becomes (3×6 + 1×4) ÷ 24, or (18 + 4) ÷ 24, which equals 22/24. Simplifying by the GCD (2), the final result is 11/12.
How to add and subtract fractions
Fractions can only be added or subtracted directly when they share the same denominator. When the denominators differ, the simplest approach is to multiply one denominator by the other to get a common denominator, adjusting each numerator by the same proportion before adding or subtracting.
How to multiply and divide fractions
Multiplying fractions is straightforward: multiply numerator by numerator and denominator by denominator. Dividing fractions follows the flip-and-multiply rule, which in practice means swapping the numerator and denominator of the second fraction before multiplying.
Proper fractions, improper fractions, and mixed numbers
A fraction is called proper when the numerator is smaller than the denominator, like 3/4, representing a value less than 1. When the numerator is equal to or greater than the denominator, like 7/4, it is called improper and represents a value equal to or greater than 1. In those cases, the improper fraction can also be written as a mixed number, combining a whole number with a proper fraction: 7/4 is equivalent to 1 and 3/4.